%   ORIGINAL: h4/EncodeVar/fixed__width__unit
% Assm: HL_TRUTH: T
% Assm: HL_FALSITY: ~F
% Assm: HL_BOOL_CASES: !t. (t <=> T) \/ (t <=> F)
% Assm: HL_EXT: !f g. (!x. f x = g x) ==> f = g
% Assm: h4/Coder/unit__coder__def: !p. h4/Coder/unit__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__unit (h4/Decode/decode__unit p))
% Assm: h4/Coder/wf__coder__unit: !p. h4/Encode/wf__pred p ==> h4/Coder/wf__coder (h4/Coder/unit__coder p)
% Assm: h4/bool/TRUTH: T
% Assm: h4/Decode/decode__unit__def: !p. h4/Decode/decode__unit p = h4/Decode/enc2dec p h4/Encode/encode__unit
% Assm: h4/Encode/wf__encode__unit: !p. h4/Encode/wf__encoder p h4/Encode/encode__unit
% Assm: h4/bool/EQ__CLAUSES_c1: !t. (t <=> T) <=> t
% Assm: h4/bool/EQ__SYM__EQ: !y x. x = y <=> y = x
% Assm: h4/bool/REFL__CLAUSE: !x. x = x <=> T
% Assm: h4/bool/IMP__CONG: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm: h4/bool/AND__IMP__INTRO: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm: h4/EncodeVar/fixed__width__def: !n c. h4/EncodeVar/fixed__width n c <=> (!x. h4/Coder/domain c x ==> h4/list/LENGTH (h4/Coder/encoder c x) = n)
% Assm: h4/bool/AND__CLAUSES_c0: !t. T /\ t <=> t
% Assm: h4/EncodeVar/fixed__width__exists: !phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((?x. h4/Coder/domain c x /\ phi x) <=> h4/bool/RES__EXISTS (h4/EncodeVar/of__length n) (\w. phi (h4/Coder/decoder c w)))
% Assm: h4/EncodeVar/fixed__width__univ: !phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((!x. h4/Coder/domain c x ==> phi x) <=> h4/bool/RES__FORALL (h4/EncodeVar/of__length n) (\w. phi (h4/Coder/decoder c w)))
% Assm: h4/bool/IMP__CLAUSES_c1: !t. t ==> T <=> T
% Assm: h4/Coder/wf__coder__def: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm: h4/one/one__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\b. b) rep
% Assm: h4/bool/boolAxiom: !t2 t1. ?fn. fn T = t1 /\ fn F = t2
% Assm: h4/Decode/wf__decode__unit: !p. h4/Decode/wf__decoder p (h4/Decode/decode__unit p)
% Assm: h4/Decode/decode__unit0: !p l. h4/Decode/decode__unit p l = h4/bool/COND (p h4/one/one0) (h4/option/SOME (h4/pair/_2C h4/one/one0 l)) h4/option/NONE
% Assm: h4/Decode/dec2enc__decode__unit: !x p. p x ==> h4/Decode/dec2enc (h4/Decode/decode__unit p) x = h4/Encode/encode__unit x
% Assm: h4/one/one1: !v. v = h4/one/one0
% Assm: h4/bool/IMP__ANTISYM__AX: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm: h4/option/option__TY__DEF: ?rep. h4/bool/TYPE__DEFINITION (\x. T) rep
% Assm: h4/one/one__axiom: !g f. f = g
% Assm: h4/Decode/wf__enc2dec: !p e. h4/Encode/wf__encoder p e ==> h4/Decode/wf__decoder p (h4/Decode/enc2dec p e)
% Assm: h4/bool/FORALL__SIMP: !t. (!x. t) <=> t
% Assm: h4/Encode/encode__unit__def: !v0. h4/Encode/encode__unit v0 = h4/list/NIL
% Assm: h4/bool/ABS__REP__THM: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. abs (rep a) = a) /\ (!r. P r <=> rep (abs r) = r))
% Assm: h4/bool/T__DEF: T <=> (\x. x) = (\x. x)
% Assm: h4/one/one__Axiom: !e. h4/bool/_3F_21 (\fn. fn h4/one/one0 = e)
% Assm: h4/bool/FALSITY: !t. F ==> t
% Assm: h4/option/OPTION__GUARD__def_c0: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm: h4/option/OPTION__GUARD__def_c1: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm: h4/bool/EXCLUDED__MIDDLE: !t. t \/ ~t
% Assm: h4/bool/EXISTS__UNIQUE__DEF: h4/bool/_3F_21 = (\P. $exists P /\ (!x y. P x /\ P y ==> x = y))
% Assm: h4/sat/NOT__NOT: !t. ~ ~t <=> t
% Assm: h4/bool/IMP__CLAUSES_c4: !t. t ==> F <=> ~t
% Assm: h4/bool/NOT__CLAUSES_c0: !t. ~ ~t <=> t
% Assm: h4/bool/DE__MORGAN__THM_c1: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm: h4/sat/AND__INV__IMP: !A. A ==> ~A ==> F
% Assm: h4/sat/pth__ni1: !q p. ~(p ==> q) ==> p
% Assm: h4/sat/OR__DUAL2: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm: h4/sat/pth__no1: !q p. ~(p \/ q) ==> ~p
% Assm: h4/sat/dc__neg: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm: h4/bool/IMP__F: !t. (t ==> F) ==> ~t
% Assm: h4/sat/pth__nn: !p. ~ ~p ==> p
% Assm: h4/sat/pth__ni2: !q p. ~(p ==> q) ==> ~q
% Assm: h4/sat/dc__disj: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm: h4/sat/dc__eq: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm: h4/sat/OR__DUAL3: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm: h4/bool/DISJ__SYM: !B A. A \/ B <=> B \/ A
% Assm: h4/bool/F__IMP: !t. ~t ==> t ==> F
% Assm: h4/bool/DISJ__ASSOC: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm: h4/sat/dc__imp: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm: h4/sat/AND__INV2: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm: h4/sat/pth__no2: !q p. ~(p \/ q) ==> ~q
% Assm: h4/EncodeVar/of__length__def: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Assm: h4/Coder/wf__coder__closed: !c. h4/Coder/wf__coder c ==> (!l. h4/Coder/domain c (h4/Coder/decoder c l))
% Assm: h4/Coder/wf__coder0: !c. h4/Coder/wf__coder c ==> (!x. h4/Coder/domain c x ==> h4/Coder/decoder c (h4/Coder/encoder c x) = x)
% Assm: h4/bool/RIGHT__FORALL__OR__THM: !Q P. (!x. P \/ Q x) <=> P \/ (!x. Q x)
% Assm: h4/Decode/dec2enc__enc2dec: !x p e. h4/Encode/wf__encoder p e /\ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = e x
% Assm: h4/one/one__case__def: !x u. h4/one/one__CASE u x = x
% Assm: h4/one/one__induction: !P. P h4/one/one0 ==> (!x. P x)
% Assm: h4/bool/AND__CLAUSES_c1: !t. t /\ T <=> t
% Assm: h4/bool/AND__CLAUSES_c4: !t. t /\ t <=> t
% Assm: h4/rich__list/IS__PREFIX_c0: !l. h4/list/isPREFIX h4/list/NIL l <=> T
% Assm: h4/Encode/wf__encoder__def: !p e. h4/Encode/wf__encoder p e <=> (!x y. p x /\ p y /\ h4/list/isPREFIX (e y) (e x) ==> x = y)
% Assm: h4/bool/NOT__CLAUSES_c1: ~T <=> F
% Assm: h4/bool/BOOL__CASES__AX: !t. (t <=> T) \/ (t <=> F)
% Assm: h4/bool/NOT__CLAUSES_c2: ~F <=> T
% Assm: h4/option/option__REP__ABS__DEF_c1: !r. (\x. T) r <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm: h4/option/OPTION__GUARD__EQ__THM_c1: !b. h4/option/OPTION__GUARD b = h4/option/NONE <=> ~b
% Assm: h4/option/NONE__DEF: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm: h4/bool/COND__CLAUSES_c0: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm: h4/bool/COND__CLAUSES_c1: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm: h4/option/OPTION__GUARD__EQ__THM_c0: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% Assm: h4/option/OPTION__GUARD__COND: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm: h4/option/SOME__DEF: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm: h4/option/option__REP__ABS__DEF_c0: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm: h4/bool/RES__FORALL__THM: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> f x)
% Assm: h4/basicSize/one__size__def: !x. h4/basicSize/one__size x = h4/num/0
% Assm: h4/bool/ETA__AX: !t. (\x. t x) = t
% Assm: h4/one/one__case__thm: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm: h4/one/one__prim__rec: !e. ?fn. fn h4/one/one0 = e
% Assm: h4/one/FORALL__ONE: !P. (!x. P x) <=> P h4/one/one0
% Assm: h4/one/one__DEF: h4/one/one0 = h4/min/_40 (\x. T)
% Assm: h4/Decode/enc2dec__some: !x t p l e. h4/Encode/wf__encoder p e ==> (h4/Decode/enc2dec p e l = h4/option/SOME (h4/pair/_2C x t) <=> p x /\ l = h4/list/APPEND (e x) t)
% Assm: h4/Decode/enc2dec__none: !p l e. h4/Decode/enc2dec p e l = h4/option/NONE <=> (!x t. p x ==> ~(l = h4/list/APPEND (e x) t))
% Assm: h4/list/APPEND0_c0: !l. h4/list/APPEND h4/list/NIL l = l
% Assm: h4/bool/IMP__CLAUSES_c2: !t. F ==> t <=> T
% Assm: h4/sat/dc__conj: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm: h4/bool/DE__MORGAN__THM_c0: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm: h4/bool/OR__CLAUSES_c2: !t. F \/ t <=> t
% Assm: h4/bool/EQ__CLAUSES_c3: !t. (t <=> F) <=> ~t
% Assm: h4/bool/NOT__EXISTS__THM: !P. ~(?x. P x) <=> (!x. ~P x)
% Assm: h4/bool/RES__EXISTS__THM: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ f x)
% Assm: h4/bool/OR__CLAUSES_c0: !t. T \/ t <=> T
% Assm: h4/bool/TYPE__DEFINITION__THM: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. rep x_27 = rep x_27_27 ==> x_27 = x_27_27) /\ (!x. P x <=> (?x_27. x = rep x_27))
% Assm: h4/bool/EQ__CLAUSES_c0: !t. (T <=> t) <=> t
% Assm: h4/bool/EQ__CLAUSES_c2: !t. (F <=> t) <=> ~t
% Assm: h4/option/NOT__NONE__SOME: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm: h4/option/SOME__11: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm: h4/bool/EQ__IMP__THM: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm: h4/prim__rec/num__Axiom: !f e. ?fn. fn h4/num/0 = e /\ (!n. fn (h4/num/SUC n) = f n (fn n))
% Assm: h4/Decode/dec__bnum__def_c1: !m l. h4/Decode/dec__bnum (h4/num/SUC m) l = h4/list/list__CASE l h4/option/NONE (\h t. h4/option/option__CASE (h4/Decode/dec__bnum m t) h4/option/NONE (\v_27. h4/pair/pair__CASE v_27 (\n t_27. h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n) (h4/bool/COND h (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0)) t_27))))
% Assm: h4/Decode/decode__num__total: !l. h4/Decode/decode__num (h4/combin/K T) l = h4/list/list__CASE l h4/option/NONE (\v2 t_27. h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE (\v6 t. h4/bool/COND v6 (h4/option/SOME (h4/pair/_2C h4/num/0 t)) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)))))) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t_27) h4/option/NONE (\v1. h4/pair/pair__CASE v1 (\v t_270. h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)))))
% Assm: h4/arithmetic/SUB__LEFT__EQ: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/SUB__RIGHT__EQ: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm: h4/arithmetic/SUB__RIGHT__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3C_3D m (h4/arithmetic/_2B n p)
% Assm: h4/arithmetic/DIV__UNIQUE: !q n k. (?r. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n) ==> h4/arithmetic/DIV k n = q
% Assm: h4/arithmetic/SUB__LEFT__ADD: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm: h4/arithmetic/NOT__NUM__EQ: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/MULT__DIV: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm: h4/arithmetic/SUC__ONE__ADD: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm: h4/arithmetic/NOT__LEQ: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm: h4/arithmetic/MULT__EQ__0: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm: h4/arithmetic/ADD__SUB: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm: h4/arithmetic/EVEN__DOUBLE: !n. h4/arithmetic/EVEN (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm: h4/arithmetic/ZERO__LESS__EQ: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm: h4/arithmetic/LESS__EQ__TRANS: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm: h4/arithmetic/MULT__CLAUSES_c2: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm: h4/arithmetic/MULT__COMM: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm: h4/bool/BOOL__EQ__DISTINCT_c0: ~(T <=> F)
% Assm: h4/numeral/numeral__suc_c0: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm: h4/arithmetic/ADD__MONO__LESS__EQ: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm: h4/arithmetic/ADD__EQ__0: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm: h4/arithmetic/ADD1: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Goal: !p. h4/EncodeVar/fixed__width h4/num/0 (h4/Coder/unit__coder p)
%   PROCESSED
% Assm [HLu_TRUTH]: T
% Assm [HLu_FALSITY]: ~F
% Assm [HLu_BOOLu_CASES]: !t. (t <=> T) \/ (t <=> F)
% Assm [HLu_EXT]: !f g. (!x. happ f x = happ g x) ==> f = g
% Assm [h4s_Coders_unitu_u_coderu_u_def]: !p. h4/Coder/unit__coder p = h4/pair/_2C p (h4/pair/_2C h4/Encode/encode__unit (h4/Decode/decode__unit p))
% Assm [h4s_Coders_wfu_u_coderu_u_unit]: !p. h4/Encode/wf__pred p ==> h4/Coder/wf__coder (h4/Coder/unit__coder p)
% Assm [h4s_bools_TRUTH]: T
% Assm [h4s_Decodes_decodeu_u_unitu_u_def]: !p. h4/Decode/decode__unit p = h4/Decode/enc2dec p h4/Encode/encode__unit
% Assm [h4s_Encodes_wfu_u_encodeu_u_unit]: !p. h4/Encode/wf__encoder p h4/Encode/encode__unit
% Assm [h4s_bools_EQu_u_CLAUSESu_c1]: !t. (t <=> T) <=> t
% Assm [h4s_bools_EQu_u_SYMu_u_EQ]: !y x. x = y <=> y = x
% Assm [h4s_bools_REFLu_u_CLAUSE]: !x. x = x <=> T
% Assm [h4s_bools_IMPu_u_CONG]: !y_27 y x_27 x. (x <=> x_27) /\ (x_27 ==> (y <=> y_27)) ==> (x ==> y <=> x_27 ==> y_27)
% Assm [h4s_bools_ANDu_u_IMPu_u_INTRO]: !t3 t2 t1. t1 ==> t2 ==> t3 <=> t1 /\ t2 ==> t3
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_def]: !n c. h4/EncodeVar/fixed__width n c <=> (!x. h4/Coder/domain c x ==> h4/list/LENGTH (h4/Coder/encoder c x) = n)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c0]: !t. T /\ t <=> t
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_exists]: !_0. (!phi c w. happ (happ (happ _0 phi) c) w <=> happ phi (h4/Coder/decoder c w)) ==> (!phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((?x. h4/Coder/domain c x /\ happ phi x) <=> h4/bool/RES__EXISTS (h4/EncodeVar/of__length n) (happ (happ _0 phi) c)))
% Assm [h4s_EncodeVars_fixedu_u_widthu_u_univ]: !_0. (!phi c w. happ (happ (happ _0 phi) c) w <=> happ phi (h4/Coder/decoder c w)) ==> (!phi n c. h4/Coder/wf__coder c /\ h4/EncodeVar/fixed__width n c ==> ((!x. h4/Coder/domain c x ==> happ phi x) <=> h4/bool/RES__FORALL (h4/EncodeVar/of__length n) (happ (happ _0 phi) c)))
% Assm [h4s_bools_IMPu_u_CLAUSESu_c1]: !t. t ==> T <=> T
% Assm [h4s_Coders_wfu_u_coderu_u_def]: !p e d. h4/Coder/wf__coder (h4/pair/_2C p (h4/pair/_2C e d)) <=> h4/Encode/wf__pred p /\ h4/Encode/wf__encoder p e /\ d = h4/Decode/enc2dec p e
% Assm [h4s_ones_oneu_u_TYu_u_DEF]: !_0. (!b. happ _0 b <=> b) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_bools_boolAxiom]: !t2 t1. ?fn. happ fn T = t1 /\ happ fn F = t2
% Assm [h4s_Decodes_wfu_u_decodeu_u_unit]: !p. h4/Decode/wf__decoder p (h4/Decode/decode__unit p)
% Assm [h4s_Decodes_decodeu_u_unit0]: !p l. happ (h4/Decode/decode__unit p) l = h4/bool/COND (happ p h4/one/one0) (h4/option/SOME (h4/pair/_2C h4/one/one0 l)) h4/option/NONE
% Assm [h4s_Decodes_dec2encu_u_decodeu_u_unit]: !x p. happ p x ==> h4/Decode/dec2enc (h4/Decode/decode__unit p) x = happ h4/Encode/encode__unit x
% Assm [h4s_ones_one1]: !v. v = h4/one/one0
% Assm [h4s_bools_IMPu_u_ANTISYMu_u_AX]: !t2 t1. (t1 ==> t2) ==> (t2 ==> t1) ==> (t1 <=> t2)
% Assm [h4s_options_optionu_u_TYu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> (?rep. h4/bool/TYPE__DEFINITION _0 rep)
% Assm [h4s_ones_oneu_u_axiom]: !g f. f = g
% Assm [h4s_Decodes_wfu_u_enc2dec]: !p e. h4/Encode/wf__encoder p e ==> h4/Decode/wf__decoder p (h4/Decode/enc2dec p e)
% Assm [h4s_bools_FORALLu_u_SIMP]: !t. (!x. t) <=> t
% Assm [h4s_Encodes_encodeu_u_unitu_u_def]: !v0. happ h4/Encode/encode__unit v0 = h4/list/NIL
% Assm [h4s_bools_ABSu_u_REPu_u_THM]: !P. (?rep. h4/bool/TYPE__DEFINITION P rep) ==> (?rep abs. (!a. happ abs (happ rep a) = a) /\ (!r. happ P r <=> happ rep (happ abs r) = r))
% Assm [h4s_bools_Tu_u_DEF]: T <=> (!x. x <=> x)
% Assm [h4s_ones_oneu_u_Axiom]: !_0. (!e fn. happ (happ _0 e) fn <=> happ fn h4/one/one0 = e) ==> (!e. h4/bool/_3F_21 (happ _0 e))
% Assm [h4s_bools_FALSITY]: !t. F ==> t
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c0]: h4/option/OPTION__GUARD T = h4/option/SOME h4/one/one0
% Assm [h4s_options_OPTIONu_u_GUARDu_u_defu_c1]: h4/option/OPTION__GUARD F = h4/option/NONE
% Assm [h4s_bools_EXCLUDEDu_u_MIDDLE]: !t. t \/ ~t
% Assm [h4s_bools_EXISTSu_u_UNIQUEu_u_DEF]: !x. h4/bool/_3F_21 x <=> $exists x /\ (!x y. happ x x /\ happ x y ==> x = y)
% Assm [h4s_sats_NOTu_u_NOT]: !t. ~ ~t <=> t
% Assm [h4s_bools_IMPu_u_CLAUSESu_c4]: !t. t ==> F <=> ~t
% Assm [h4s_bools_NOTu_u_CLAUSESu_c0]: !t. ~ ~t <=> t
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c1]: !B A. ~(A \/ B) <=> ~A /\ ~B
% Assm [h4s_sats_ANDu_u_INVu_u_IMP]: !A. A ==> ~A ==> F
% Assm [h4s_sats_pthu_u_ni1]: !q p. ~(p ==> q) ==> p
% Assm [h4s_sats_ORu_u_DUAL2]: !B A. ~(A \/ B) ==> F <=> (A ==> F) ==> ~B ==> F
% Assm [h4s_sats_pthu_u_no1]: !q p. ~(p \/ q) ==> ~p
% Assm [h4s_sats_dcu_u_neg]: !q p. (p <=> ~q) <=> (p \/ q) /\ (~q \/ ~p)
% Assm [h4s_bools_IMPu_u_F]: !t. (t ==> F) ==> ~t
% Assm [h4s_sats_pthu_u_nn]: !p. ~ ~p ==> p
% Assm [h4s_sats_pthu_u_ni2]: !q p. ~(p ==> q) ==> ~q
% Assm [h4s_sats_dcu_u_disj]: !r q p. (p <=> q \/ r) <=> (p \/ ~q) /\ (p \/ ~r) /\ (q \/ r \/ ~p)
% Assm [h4s_sats_dcu_u_eq]: !r q p. (p <=> q <=> r) <=> (p \/ q \/ r) /\ (p \/ ~r \/ ~q) /\ (q \/ ~r \/ ~p) /\ (r \/ ~q \/ ~p)
% Assm [h4s_sats_ORu_u_DUAL3]: !B A. ~(~A \/ B) ==> F <=> A ==> ~B ==> F
% Assm [h4s_bools_DISJu_u_SYM]: !B A. A \/ B <=> B \/ A
% Assm [h4s_bools_Fu_u_IMP]: !t. ~t ==> t ==> F
% Assm [h4s_bools_DISJu_u_ASSOC]: !C B A. A \/ B \/ C <=> (A \/ B) \/ C
% Assm [h4s_sats_dcu_u_imp]: !r q p. (p <=> q ==> r) <=> (p \/ q) /\ (p \/ ~r) /\ (~q \/ r \/ ~p)
% Assm [h4s_sats_ANDu_u_INV2]: !A. (~A ==> F) ==> (A ==> F) ==> F
% Assm [h4s_sats_pthu_u_no2]: !q p. ~(p \/ q) ==> ~q
% Assm [h4s_EncodeVars_ofu_u_lengthu_u_def]: !n l. h4/bool/IN l (h4/EncodeVar/of__length n) <=> h4/list/LENGTH l = n
% Assm [h4s_Coders_wfu_u_coderu_u_closed]: !c. h4/Coder/wf__coder c ==> (!l. h4/Coder/domain c (h4/Coder/decoder c l))
% Assm [h4s_Coders_wfu_u_coder0]: !c. h4/Coder/wf__coder c ==> (!x. h4/Coder/domain c x ==> h4/Coder/decoder c (h4/Coder/encoder c x) = x)
% Assm [h4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM]: !Q P. (!x. P \/ happ Q x) <=> P \/ (!x. happ Q x)
% Assm [h4s_Decodes_dec2encu_u_enc2dec]: !x p e. h4/Encode/wf__encoder p e /\ happ p x ==> h4/Decode/dec2enc (h4/Decode/enc2dec p e) x = happ e x
% Assm [h4s_ones_oneu_u_caseu_u_def]: !x u. h4/one/one__CASE u x = x
% Assm [h4s_ones_oneu_u_induction]: !P. happ P h4/one/one0 ==> (!x. happ P x)
% Assm [h4s_bools_ANDu_u_CLAUSESu_c1]: !t. t /\ T <=> t
% Assm [h4s_bools_ANDu_u_CLAUSESu_c4]: !t. t /\ t <=> t
% Assm [h4s_richu_u_lists_ISu_u_PREFIXu_c0]: !l. h4/list/isPREFIX h4/list/NIL l <=> T
% Assm [h4s_Encodes_wfu_u_encoderu_u_def]: !p e. h4/Encode/wf__encoder p e <=> (!x y. happ p x /\ happ p y /\ h4/list/isPREFIX (happ e y) (happ e x) ==> x = y)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c1]: ~T <=> F
% Assm [h4s_bools_BOOLu_u_CASESu_u_AX]: !t. (t <=> T) \/ (t <=> F)
% Assm [h4s_bools_NOTu_u_CLAUSESu_c2]: ~F <=> T
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1]: !r. T <=> h4/option/option__REP (h4/option/option__ABS r) = r
% Assm [h4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c1]: !b. h4/option/OPTION__GUARD b = h4/option/NONE <=> ~b
% Assm [h4s_options_NONEu_u_DEF]: h4/option/NONE = h4/option/option__ABS (h4/sum/INR h4/one/one0)
% Assm [h4s_bools_CONDu_u_CLAUSESu_c0]: !t2 t1. h4/bool/COND T t1 t2 = t1
% Assm [h4s_bools_CONDu_u_CLAUSESu_c1]: !t2 t1. h4/bool/COND F t1 t2 = t2
% Assm [h4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0]: !b. h4/option/OPTION__GUARD b = h4/option/SOME h4/one/one0 <=> b
% Assm [h4s_options_OPTIONu_u_GUARDu_u_COND]: !b. h4/option/OPTION__GUARD b = h4/bool/COND b (h4/option/SOME h4/one/one0) h4/option/NONE
% Assm [h4s_options_SOMEu_u_DEF]: !x. h4/option/SOME x = h4/option/option__ABS (h4/sum/INL x)
% Assm [h4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0]: !a. h4/option/option__ABS (h4/option/option__REP a) = a
% Assm [h4s_bools_RESu_u_FORALLu_u_THM]: !f P. h4/bool/RES__FORALL P f <=> (!x. h4/bool/IN x P ==> happ f x)
% Assm [h4s_basicSizes_oneu_u_sizeu_u_def]: !x. h4/basicSize/one__size x = h4/num/0
% Assm [h4s_bools_ETAu_u_AX]: !t x. happ t x = happ t x
% Assm [h4s_ones_oneu_u_caseu_u_thm]: !x. h4/one/one__CASE h4/one/one0 x = x
% Assm [h4s_ones_oneu_u_primu_u_rec]: !e. ?fn. happ fn h4/one/one0 = e
% Assm [h4s_ones_FORALLu_u_ONE]: !P. (!x. happ P x) <=> happ P h4/one/one0
% Assm [h4s_ones_oneu_u_DEF]: !_0. (!x. happ _0 x <=> T) ==> h4/one/one0 = h4/min/_40 _0
% Assm [h4s_Decodes_enc2decu_u_some]: !x t p l e. h4/Encode/wf__encoder p e ==> (happ (h4/Decode/enc2dec p e) l = h4/option/SOME (h4/pair/_2C x t) <=> happ p x /\ l = h4/list/APPEND (happ e x) t)
% Assm [h4s_Decodes_enc2decu_u_none]: !p l e. happ (h4/Decode/enc2dec p e) l = h4/option/NONE <=> (!x t. happ p x ==> ~(l = h4/list/APPEND (happ e x) t))
% Assm [h4s_lists_APPEND0u_c0]: !l. h4/list/APPEND h4/list/NIL l = l
% Assm [h4s_bools_IMPu_u_CLAUSESu_c2]: !t. F ==> t <=> T
% Assm [h4s_sats_dcu_u_conj]: !r q p. (p <=> q /\ r) <=> (p \/ ~q \/ ~r) /\ (q \/ ~p) /\ (r \/ ~p)
% Assm [h4s_bools_DEu_u_MORGANu_u_THMu_c0]: !B A. ~(A /\ B) <=> ~A \/ ~B
% Assm [h4s_bools_ORu_u_CLAUSESu_c2]: !t. F \/ t <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c3]: !t. (t <=> F) <=> ~t
% Assm [h4s_bools_NOTu_u_EXISTSu_u_THM]: !P. ~(?x. happ P x) <=> (!x. ~happ P x)
% Assm [h4s_bools_RESu_u_EXISTSu_u_THM]: !f P. h4/bool/RES__EXISTS P f <=> (?x. h4/bool/IN x P /\ happ f x)
% Assm [h4s_bools_ORu_u_CLAUSESu_c0]: !t. T \/ t <=> T
% Assm [h4s_bools_TYPEu_u_DEFINITIONu_u_THM]: !rep P. h4/bool/TYPE__DEFINITION P rep <=> (!x_27 x_27_27. happ rep x_27 = happ rep x_27_27 ==> x_27 = x_27_27) /\ (!x. happ P x <=> (?x_27. x = happ rep x_27))
% Assm [h4s_bools_EQu_u_CLAUSESu_c0]: !t. (T <=> t) <=> t
% Assm [h4s_bools_EQu_u_CLAUSESu_c2]: !t. (F <=> t) <=> ~t
% Assm [h4s_options_NOTu_u_NONEu_u_SOME]: !x. ~(h4/option/NONE = h4/option/SOME x)
% Assm [h4s_options_SOMEu_u_11]: !y x. h4/option/SOME x = h4/option/SOME y <=> x = y
% Assm [h4s_bools_EQu_u_IMPu_u_THM]: !t2 t1. (t1 <=> t2) <=> (t1 ==> t2) /\ (t2 ==> t1)
% Assm [h4s_primu_u_recs_numu_u_Axiom]: !f e. ?fn. happ fn h4/num/0 = e /\ (!n. happ fn (h4/num/SUC n) = happ (happ f n) (happ fn n))
% Assm [h4s_Decodes_decu_u_bnumu_u_defu_c1]: !_4. (!n h t_27. happ (happ (happ _4 n) h) t_27 = h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n) (h4/bool/COND h (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) h4/num/0)) t_27)) ==> (!_3. (!h n. happ (happ _3 h) n = happ (happ _4 n) h) ==> (!_2. (!h v_27. happ (happ _2 h) v_27 = h4/pair/pair__CASE v_27 (happ _3 h)) ==> (!_1. (!m h t. happ (happ (happ _1 m) h) t = h4/option/option__CASE (h4/Decode/dec__bnum m t) h4/option/NONE (happ _2 h)) ==> (!_0. (!m h. happ (happ _0 m) h = happ (happ _1 m) h) ==> (!m l. h4/Decode/dec__bnum (h4/num/SUC m) l = h4/list/list__CASE l h4/option/NONE (happ _0 m))))))
% Assm [h4s_Decodes_decodeu_u_numu_u_total]: !_9. (!v t_270. happ (happ _9 v) t_270 = h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO))) t_270)) ==> (!_8. (!v. happ _8 v = happ _9 v) ==> (!_7. (!v1. happ _7 v1 = h4/pair/pair__CASE v1 _8) ==> (!_6. (!v t_270. happ (happ _6 v) t_270 = h4/option/SOME (h4/pair/_2C (h4/arithmetic/_2B (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) v) (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))) t_270)) ==> (!_5. (!v. happ _5 v = happ _6 v) ==> (!_4. (!v1. happ _4 v1 = h4/pair/pair__CASE v1 _5) ==> (!_3. (!v6 t. happ (happ _3 v6) t = h4/bool/COND v6 (h4/option/SOME (h4/pair/_2C h4/num/0 t)) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t) h4/option/NONE _4)) ==> (!_2. (!v6. happ _2 v6 = happ _3 v6) ==> (!_1. (!v2 t_27. happ (happ _1 v2) t_27 = h4/bool/COND v2 (h4/list/list__CASE t_27 h4/option/NONE _2) (h4/option/option__CASE (h4/Decode/decode__num (h4/combin/K T) t_27) h4/option/NONE _7)) ==> (!_0. (!v2. happ _0 v2 = happ _1 v2) ==> (!l. h4/Decode/decode__num (h4/combin/K T) l = h4/list/list__CASE l h4/option/NONE _0))))))))))
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_EQ]: !p n m. m = h4/arithmetic/_2D n p <=> h4/arithmetic/_2B m p = n \/ h4/arithmetic/_3C_3D m h4/num/0 /\ h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_EQ]: !p n m. h4/arithmetic/_2D m n = p <=> m = h4/arithmetic/_2B n p \/ h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D p h4/num/0
% Assm [h4s_arithmetics_SUBu_u_RIGHTu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2D m n) p <=> h4/arithmetic/_3C_3D m (h4/arithmetic/_2B n p)
% Assm [h4s_arithmetics_DIVu_u_UNIQUE]: !q n k. (?r. k = h4/arithmetic/_2B (h4/arithmetic/_2A q n) r /\ h4/prim__rec/_3C r n) ==> h4/arithmetic/DIV k n = q
% Assm [h4s_arithmetics_SUBu_u_LEFTu_u_ADD]: !p n m. h4/arithmetic/_2B m (h4/arithmetic/_2D n p) = h4/bool/COND (h4/arithmetic/_3C_3D n p) m (h4/arithmetic/_2D (h4/arithmetic/_2B m n) p)
% Assm [h4s_arithmetics_NOTu_u_NUMu_u_EQ]: !n m. ~(m = n) <=> h4/arithmetic/_3C_3D (h4/num/SUC m) n \/ h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_MULTu_u_DIV]: !q n. h4/prim__rec/_3C h4/num/0 n ==> h4/arithmetic/DIV (h4/arithmetic/_2A q n) n = q
% Assm [h4s_arithmetics_SUCu_u_ONEu_u_ADD]: !n. h4/num/SUC n = h4/arithmetic/_2B (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) n
% Assm [h4s_arithmetics_NOTu_u_LEQ]: !n m. ~h4/arithmetic/_3C_3D m n <=> h4/arithmetic/_3C_3D (h4/num/SUC n) m
% Assm [h4s_arithmetics_MULTu_u_EQu_u_0]: !n m. h4/arithmetic/_2A m n = h4/num/0 <=> m = h4/num/0 \/ n = h4/num/0
% Assm [h4s_arithmetics_ADDu_u_SUB]: !c a. h4/arithmetic/_2D (h4/arithmetic/_2B a c) c = a
% Assm [h4s_arithmetics_EVENu_u_DOUBLE]: !n. h4/arithmetic/EVEN (h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT2 h4/arithmetic/ZERO)) n)
% Assm [h4s_arithmetics_ZEROu_u_LESSu_u_EQ]: !n. h4/arithmetic/_3C_3D h4/num/0 n
% Assm [h4s_arithmetics_LESSu_u_EQu_u_TRANS]: !p n m. h4/arithmetic/_3C_3D m n /\ h4/arithmetic/_3C_3D n p ==> h4/arithmetic/_3C_3D m p
% Assm [h4s_arithmetics_MULTu_u_CLAUSESu_c2]: !m. h4/arithmetic/_2A (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO)) m = m
% Assm [h4s_arithmetics_MULTu_u_COMM]: !n m. h4/arithmetic/_2A m n = h4/arithmetic/_2A n m
% Assm [h4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0]: ~(T <=> F)
% Assm [h4s_numerals_numeralu_u_sucu_c0]: h4/num/SUC h4/arithmetic/ZERO = h4/arithmetic/BIT1 h4/arithmetic/ZERO
% Assm [h4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ]: !p n m. h4/arithmetic/_3C_3D (h4/arithmetic/_2B m n) (h4/arithmetic/_2B m p) <=> h4/arithmetic/_3C_3D n p
% Assm [h4s_arithmetics_ADDu_u_EQu_u_0]: !n m. h4/arithmetic/_2B m n = h4/num/0 <=> m = h4/num/0 /\ n = h4/num/0
% Assm [h4s_arithmetics_ADD1]: !m. h4/num/SUC m = h4/arithmetic/_2B m (h4/arithmetic/NUMERAL (h4/arithmetic/BIT1 h4/arithmetic/ZERO))
% Goal: !p. h4/EncodeVar/fixed__width h4/num/0 (h4/Coder/unit__coder p)
fof(aHLu_TRUTH, axiom, p(s(t_bool,t))).
fof(aHLu_FALSITY, axiom, ~ (p(s(t_bool,f)))).
fof(aHLu_BOOLu_CASES, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(aHLu_EXT, axiom, ![TV_Q1147920,TV_Q1147916]: ![V_f, V_g]: (![V_x]: s(TV_Q1147916,happ(s(t_fun(TV_Q1147920,TV_Q1147916),V_f),s(TV_Q1147920,V_x))) = s(TV_Q1147916,happ(s(t_fun(TV_Q1147920,TV_Q1147916),V_g),s(TV_Q1147920,V_x))) => s(t_fun(TV_Q1147920,TV_Q1147916),V_f) = s(t_fun(TV_Q1147920,TV_Q1147916),V_g))).
fof(ah4s_Coders_unitu_u_coderu_u_def, axiom, ![V_p]: s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_coders_unitu_u_coder(s(t_fun(t_h4s_ones_one,t_bool),V_p))) = s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p)))))))).
fof(ah4s_Coders_wfu_u_coderu_u_unit, axiom, ![V_p]: (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(t_h4s_ones_one,t_bool),V_p)))) => p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_coders_unitu_u_coder(s(t_fun(t_h4s_ones_one,t_bool),V_p)))))))).
fof(ah4s_bools_TRUTH, axiom, p(s(t_bool,t))).
fof(ah4s_Decodes_decodeu_u_unitu_u_def, axiom, ![V_p]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit)))).
fof(ah4s_Encodes_wfu_u_encodeu_u_unit, axiom, ![V_p]: p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c1, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_SYMu_u_EQ, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_y) <=> s(TV_u_27a,V_y) = s(TV_u_27a,V_x))).
fof(ah4s_bools_REFLu_u_CLAUSE, axiom, ![TV_u_27a]: ![V_x]: (s(TV_u_27a,V_x) = s(TV_u_27a,V_x) <=> p(s(t_bool,t)))).
fof(ah4s_bools_IMPu_u_CONG, axiom, ![V_yu_27, V_y, V_xu_27, V_x]: ((s(t_bool,V_x) = s(t_bool,V_xu_27) & (p(s(t_bool,V_xu_27)) => s(t_bool,V_y) = s(t_bool,V_yu_27))) => ((p(s(t_bool,V_x)) => p(s(t_bool,V_y))) <=> (p(s(t_bool,V_xu_27)) => p(s(t_bool,V_yu_27)))))).
fof(ah4s_bools_ANDu_u_IMPu_u_INTRO, axiom, ![V_t3, V_t2, V_t1]: ((p(s(t_bool,V_t1)) => (p(s(t_bool,V_t2)) => p(s(t_bool,V_t3)))) <=> ((p(s(t_bool,V_t1)) & p(s(t_bool,V_t2))) => p(s(t_bool,V_t3))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_def, axiom, ![TV_u_27a]: ![V_n, V_c]: (p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) <=> ![V_x]: (p(s(t_bool,h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x)))) => s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(t_bool),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x))))) = s(t_h4s_nums_num,V_n)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_exists, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_phi, V_c, V_w]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(t_h4s_lists_list(t_bool),V_w))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_w))))) => ![V_phi, V_n, V_c]: ((p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))))) => (?[V_x]: (p(s(t_bool,h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,V_x))))) <=> p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(t_h4s_lists_list(t_bool),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))))))))).
fof(ah4s_EncodeVars_fixedu_u_widthu_u_univ, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_phi, V_c, V_w]: s(t_bool,happ(s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))),s(t_h4s_lists_list(t_bool),V_w))) = s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_w))))) => ![V_phi, V_n, V_c]: ((p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) & p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,V_n),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c))))) => (![V_x]: (p(s(t_bool,h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_phi),s(TV_u_27a,V_x))))) <=> p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(t_h4s_lists_list(t_bool),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n))),s(t_fun(t_h4s_lists_list(t_bool),t_bool),happ(s(t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool)),happ(s(t_fun(t_fun(TV_u_27a,t_bool),t_fun(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),t_fun(t_h4s_lists_list(t_bool),t_bool))),V_uu_0),s(t_fun(TV_u_27a,t_bool),V_phi))),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))))))))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,t))) <=> p(s(t_bool,t)))).
fof(ah4s_Coders_wfu_u_coderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e, V_d]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_bool),V_p),s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))))),h4s_pairs_u_2c(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d)))))))) <=> (p(s(t_bool,h4s_encodes_wfu_u_pred(s(t_fun(TV_u_27a,t_bool),V_p)))) & (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),V_d) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))))))).
fof(ah4s_ones_oneu_u_TYu_u_DEF, axiom, ![V_uu_0]: (![V_b]: s(t_bool,happ(s(t_fun(t_bool,t_bool),V_uu_0),s(t_bool,V_b))) = s(t_bool,V_b) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_bool,t_bool),V_uu_0),s(t_fun(t_h4s_ones_one,t_bool),V_rep)))))).
fof(ah4s_bools_boolAxiom, axiom, ![TV_u_27a]: ![V_t2, V_t1]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,t))) = s(TV_u_27a,V_t1) & s(TV_u_27a,happ(s(t_fun(t_bool,TV_u_27a),V_fn),s(t_bool,f))) = s(TV_u_27a,V_t2))).
fof(ah4s_Decodes_wfu_u_decodeu_u_unit, axiom, ![V_p]: p(s(t_bool,h4s_decodes_wfu_u_decoder(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p))))))).
fof(ah4s_Decodes_decodeu_u_unit0, axiom, ![V_p, V_l]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_ones_one,h4s_ones_one0),s(t_h4s_lists_list(t_bool),V_l))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool))),h4s_options_none)))).
fof(ah4s_Decodes_dec2encu_u_decodeu_u_unit, axiom, ![V_x, V_p]: (p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_p),s(t_h4s_ones_one,V_x)))) => s(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))),h4s_decodes_decodeu_u_unit(s(t_fun(t_h4s_ones_one,t_bool),V_p))),s(t_h4s_ones_one,V_x))) = s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit),s(t_h4s_ones_one,V_x))))).
fof(ah4s_ones_one1, axiom, ![V_v]: s(t_h4s_ones_one,V_v) = s(t_h4s_ones_one,h4s_ones_one0)).
fof(ah4s_bools_IMPu_u_ANTISYMu_u_AX, axiom, ![V_t2, V_t1]: ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) => ((p(s(t_bool,V_t2)) => p(s(t_bool,V_t1))) => s(t_bool,V_t1) = s(t_bool,V_t2)))).
fof(ah4s_options_optionu_u_TYu_u_DEF, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_x))) = s(t_bool,t) => ?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),t_bool),V_uu_0),s(t_fun(t_h4s_options_option(TV_u_27a),t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one)),V_rep)))))).
fof(ah4s_ones_oneu_u_axiom, axiom, ![TV_u_27a]: ![V_g, V_f]: s(t_fun(TV_u_27a,t_h4s_ones_one),V_f) = s(t_fun(TV_u_27a,t_h4s_ones_one),V_g)).
fof(ah4s_Decodes_wfu_u_enc2dec, axiom, ![TV_u_27a]: ![V_p, V_e]: (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) => p(s(t_bool,h4s_decodes_wfu_u_decoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))))))).
fof(ah4s_bools_FORALLu_u_SIMP, axiom, ![TV_u_27a]: ![V_t]: (![V_x]: p(s(t_bool,V_t)) <=> p(s(t_bool,V_t)))).
fof(ah4s_Encodes_encodeu_u_unitu_u_def, axiom, ![V_v0]: s(t_h4s_lists_list(t_bool),happ(s(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),h4s_encodes_encodeu_u_unit),s(t_h4s_ones_one,V_v0))) = s(t_h4s_lists_list(t_bool),h4s_lists_nil)).
fof(ah4s_bools_ABSu_u_REPu_u_THM, axiom, ![TV_u_27b,TV_u_27a]: ![V_P]: (?[V_rep]: p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) => ?[V_rep, V_abs]: (![V_a]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_a))))) = s(TV_u_27b,V_a) & ![V_r]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_r)))) <=> s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_abs),s(TV_u_27a,V_r))))) = s(TV_u_27a,V_r))))).
fof(ah4s_bools_Tu_u_DEF, axiom, (p(s(t_bool,t)) <=> ![V_x]: s(t_bool,V_x) = s(t_bool,V_x))).
fof(ah4s_ones_oneu_u_Axiom, axiom, ![TV_u_27a]: ![V_uu_0]: (![V_e, V_fn]: (p(s(t_bool,happ(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e))),s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn)))) <=> s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)) => ![V_e]: p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool),happ(s(t_fun(TV_u_27a,t_fun(t_fun(t_h4s_ones_one,TV_u_27a),t_bool)),V_uu_0),s(TV_u_27a,V_e)))))))).
fof(ah4s_bools_FALSITY, axiom, ![V_t]: (p(s(t_bool,f)) => p(s(t_bool,V_t)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c0, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,t))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_defu_c1, axiom, s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,f))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)).
fof(ah4s_bools_EXCLUDEDu_u_MIDDLE, axiom, ![V_t]: (p(s(t_bool,V_t)) | ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_EXISTSu_u_UNIQUEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: (p(s(t_bool,h4s_bools_u_3fu_21(s(t_fun(TV_u_27a,t_bool),V_x)))) <=> (p(s(t_bool,d_exists(s(t_fun(TV_u_27a,t_bool),V_x)))) & ![V_x0, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_x0)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_x),s(TV_u_27a,V_y))))) => s(TV_u_27a,V_x0) = s(TV_u_27a,V_y))))).
fof(ah4s_sats_NOTu_u_NOT, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_IMPu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c0, axiom, ![V_t]: (~ (~ (p(s(t_bool,V_t)))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c1, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) & ~ (p(s(t_bool,V_B)))))).
fof(ah4s_sats_ANDu_u_INVu_u_IMP, axiom, ![V_A]: (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_A))) => p(s(t_bool,f))))).
fof(ah4s_sats_pthu_u_ni1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_ORu_u_DUAL2, axiom, ![V_B, V_A]: ((~ ((p(s(t_bool,V_A)) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_sats_pthu_u_no1, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_p))))).
fof(ah4s_sats_dcu_u_neg, axiom, ![V_q, V_p]: ((p(s(t_bool,V_p)) <=> ~ (p(s(t_bool,V_q)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p))))))).
fof(ah4s_bools_IMPu_u_F, axiom, ![V_t]: ((p(s(t_bool,V_t)) => p(s(t_bool,f))) => ~ (p(s(t_bool,V_t))))).
fof(ah4s_sats_pthu_u_nn, axiom, ![V_p]: (~ (~ (p(s(t_bool,V_p)))) => p(s(t_bool,V_p)))).
fof(ah4s_sats_pthu_u_ni2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) => p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_sats_dcu_u_disj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_q)))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (p(s(t_bool,V_q)) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_dcu_u_eq, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> s(t_bool,V_q) = s(t_bool,V_r)) <=> ((p(s(t_bool,V_p)) | (p(s(t_bool,V_q)) | p(s(t_bool,V_r)))) & ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_q))))) & ((p(s(t_bool,V_q)) | (~ (p(s(t_bool,V_r))) | ~ (p(s(t_bool,V_p))))) & (p(s(t_bool,V_r)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_p)))))))))).
fof(ah4s_sats_ORu_u_DUAL3, axiom, ![V_B, V_A]: ((~ ((~ (p(s(t_bool,V_A))) | p(s(t_bool,V_B)))) => p(s(t_bool,f))) <=> (p(s(t_bool,V_A)) => (~ (p(s(t_bool,V_B))) => p(s(t_bool,f)))))).
fof(ah4s_bools_DISJu_u_SYM, axiom, ![V_B, V_A]: ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) <=> (p(s(t_bool,V_B)) | p(s(t_bool,V_A))))).
fof(ah4s_bools_Fu_u_IMP, axiom, ![V_t]: (~ (p(s(t_bool,V_t))) => (p(s(t_bool,V_t)) => p(s(t_bool,f))))).
fof(ah4s_bools_DISJu_u_ASSOC, axiom, ![V_C, V_B, V_A]: ((p(s(t_bool,V_A)) | (p(s(t_bool,V_B)) | p(s(t_bool,V_C)))) <=> ((p(s(t_bool,V_A)) | p(s(t_bool,V_B))) | p(s(t_bool,V_C))))).
fof(ah4s_sats_dcu_u_imp, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) => p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | p(s(t_bool,V_q))) & ((p(s(t_bool,V_p)) | ~ (p(s(t_bool,V_r)))) & (~ (p(s(t_bool,V_q))) | (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p))))))))).
fof(ah4s_sats_ANDu_u_INV2, axiom, ![V_A]: ((~ (p(s(t_bool,V_A))) => p(s(t_bool,f))) => ((p(s(t_bool,V_A)) => p(s(t_bool,f))) => p(s(t_bool,f))))).
fof(ah4s_sats_pthu_u_no2, axiom, ![V_q, V_p]: (~ ((p(s(t_bool,V_p)) | p(s(t_bool,V_q)))) => ~ (p(s(t_bool,V_q))))).
fof(ah4s_EncodeVars_ofu_u_lengthu_u_def, axiom, ![TV_u_27a]: ![V_n, V_l]: (p(s(t_bool,h4s_bools_in(s(t_h4s_lists_list(TV_u_27a),V_l),s(t_fun(t_h4s_lists_list(TV_u_27a),t_bool),h4s_encodevars_ofu_u_length(s(t_h4s_nums_num,V_n)))))) <=> s(t_h4s_nums_num,h4s_lists_length(s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_nums_num,V_n))).
fof(ah4s_Coders_wfu_u_coderu_u_closed, axiom, ![TV_u_27a]: ![V_c]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) => ![V_l]: p(s(t_bool,h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),V_l)))))))).
fof(ah4s_Coders_wfu_u_coder0, axiom, ![TV_u_27a]: ![V_c]: (p(s(t_bool,h4s_coders_wfu_u_coder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c)))) => ![V_x]: (p(s(t_bool,h4s_coders_domain(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x)))) => s(TV_u_27a,h4s_coders_decoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(t_h4s_lists_list(t_bool),h4s_coders_encoder(s(t_h4s_pairs_prod(t_fun(TV_u_27a,t_bool),t_h4s_pairs_prod(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))))),V_c),s(TV_u_27a,V_x))))) = s(TV_u_27a,V_x)))).
fof(ah4s_bools_RIGHTu_u_FORALLu_u_ORu_u_THM, axiom, ![TV_u_27a]: ![V_Q, V_P]: (![V_x]: (p(s(t_bool,V_P)) | p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))) <=> (p(s(t_bool,V_P)) | ![V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_Q),s(TV_u_27a,V_x))))))).
fof(ah4s_Decodes_dec2encu_u_enc2dec, axiom, ![TV_u_27a]: ![V_x, V_p, V_e]: ((p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x))))) => s(t_h4s_lists_list(t_bool),h4s_decodes_dec2enc(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(TV_u_27a,V_x))) = s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))))).
fof(ah4s_ones_oneu_u_caseu_u_def, axiom, ![TV_u_27a]: ![V_x, V_u]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,V_u),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_ones_oneu_u_induction, axiom, ![V_P]: (p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))) => ![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c1, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_ANDu_u_CLAUSESu_c4, axiom, ![V_t]: ((p(s(t_bool,V_t)) & p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_richu_u_lists_ISu_u_PREFIXu_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_bool,h4s_lists_isprefix(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_bool,t)).
fof(ah4s_Encodes_wfu_u_encoderu_u_def, axiom, ![TV_u_27a]: ![V_p, V_e]: (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) <=> ![V_x, V_y]: ((p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_y)))) & p(s(t_bool,h4s_lists_isprefix(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_y))),s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x)))))))) => s(TV_u_27a,V_x) = s(TV_u_27a,V_y)))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c1, axiom, (~ (p(s(t_bool,t))) <=> p(s(t_bool,f)))).
fof(ah4s_bools_BOOLu_u_CASESu_u_AX, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,t) | s(t_bool,V_t) = s(t_bool,f))).
fof(ah4s_bools_NOTu_u_CLAUSESu_c2, axiom, (~ (p(s(t_bool,f))) <=> p(s(t_bool,t)))).
fof(ah4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c1, axiom, ![TV_u_27a]: ![V_r]: (p(s(t_bool,t)) <=> s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))))) = s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),V_r))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c1, axiom, ![V_b]: (s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none) <=> ~ (p(s(t_bool,V_b))))).
fof(ah4s_options_NONEu_u_DEF, axiom, ![TV_u_27a]: s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inr(s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_bools_CONDu_u_CLAUSESu_c0, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,t),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t1)).
fof(ah4s_bools_CONDu_u_CLAUSESu_c1, axiom, ![TV_u_27a]: ![V_t2, V_t1]: s(TV_u_27a,h4s_bools_cond(s(t_bool,f),s(TV_u_27a,V_t1),s(TV_u_27a,V_t2))) = s(TV_u_27a,V_t2)).
fof(ah4s_options_OPTIONu_u_GUARDu_u_EQu_u_THMu_c0, axiom, ![V_b]: (s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))) <=> p(s(t_bool,V_b)))).
fof(ah4s_options_OPTIONu_u_GUARDu_u_COND, axiom, ![V_b]: s(t_h4s_options_option(t_h4s_ones_one),h4s_options_optionu_u_guard(s(t_bool,V_b))) = s(t_h4s_options_option(t_h4s_ones_one),h4s_bools_cond(s(t_bool,V_b),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_some(s(t_h4s_ones_one,h4s_ones_one0))),s(t_h4s_options_option(t_h4s_ones_one),h4s_options_none)))).
fof(ah4s_options_SOMEu_u_DEF, axiom, ![TV_u_27a]: ![V_x]: s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_sums_inl(s(TV_u_27a,V_x)))))).
fof(ah4s_options_optionu_u_REPu_u_ABSu_u_DEFu_c0, axiom, ![TV_u_27a]: ![V_a]: s(t_h4s_options_option(TV_u_27a),h4s_options_optionu_u_abs(s(t_h4s_sums_sum(TV_u_27a,t_h4s_ones_one),h4s_options_optionu_u_rep(s(t_h4s_options_option(TV_u_27a),V_a))))) = s(t_h4s_options_option(TV_u_27a),V_a)).
fof(ah4s_bools_RESu_u_FORALLu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_forall(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ![V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) => p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_basicSizes_oneu_u_sizeu_u_def, axiom, ![V_x]: s(t_h4s_nums_num,h4s_basicsizes_oneu_u_size(s(t_h4s_ones_one,V_x))) = s(t_h4s_nums_num,h4s_nums_0)).
fof(ah4s_bools_ETAu_u_AX, axiom, ![TV_u_27b,TV_u_27a]: ![V_t, V_x]: s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x))) = s(TV_u_27b,happ(s(t_fun(TV_u_27a,TV_u_27b),V_t),s(TV_u_27a,V_x)))).
fof(ah4s_ones_oneu_u_caseu_u_thm, axiom, ![TV_u_27a]: ![V_x]: s(TV_u_27a,h4s_ones_oneu_u_case(s(t_h4s_ones_one,h4s_ones_one0),s(TV_u_27a,V_x))) = s(TV_u_27a,V_x)).
fof(ah4s_ones_oneu_u_primu_u_rec, axiom, ![TV_u_27a]: ![V_e]: ?[V_fn]: s(TV_u_27a,happ(s(t_fun(t_h4s_ones_one,TV_u_27a),V_fn),s(t_h4s_ones_one,h4s_ones_one0))) = s(TV_u_27a,V_e)).
fof(ah4s_ones_FORALLu_u_ONE, axiom, ![V_P]: (![V_x]: p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,V_x)))) <=> p(s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_P),s(t_h4s_ones_one,h4s_ones_one0)))))).
fof(ah4s_ones_oneu_u_DEF, axiom, ![V_uu_0]: (![V_x]: s(t_bool,happ(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0),s(t_h4s_ones_one,V_x))) = s(t_bool,t) => s(t_h4s_ones_one,h4s_ones_one0) = s(t_h4s_ones_one,h4s_mins_u_40(s(t_fun(t_h4s_ones_one,t_bool),V_uu_0))))).
fof(ah4s_Decodes_enc2decu_u_some, axiom, ![TV_u_27a]: ![V_x, V_t, V_p, V_l, V_e]: (p(s(t_bool,h4s_encodes_wfu_u_encoder(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e)))) => (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(TV_u_27a,V_x),s(t_h4s_lists_list(t_bool),V_t))))) <=> (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) & s(t_h4s_lists_list(t_bool),V_l) = s(t_h4s_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_t))))))).
fof(ah4s_Decodes_enc2decu_u_none, axiom, ![TV_u_27a]: ![V_p, V_l, V_e]: (s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool)))),h4s_decodes_enc2dec(s(t_fun(TV_u_27a,t_bool),V_p),s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(TV_u_27a,t_h4s_lists_list(t_bool))),h4s_options_none) <=> ![V_x, V_t]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_p),s(TV_u_27a,V_x)))) => ~ (s(t_h4s_lists_list(t_bool),V_l) = s(t_h4s_lists_list(t_bool),h4s_lists_append(s(t_h4s_lists_list(t_bool),happ(s(t_fun(TV_u_27a,t_h4s_lists_list(t_bool)),V_e),s(TV_u_27a,V_x))),s(t_h4s_lists_list(t_bool),V_t))))))).
fof(ah4s_lists_APPEND0u_c0, axiom, ![TV_u_27a]: ![V_l]: s(t_h4s_lists_list(TV_u_27a),h4s_lists_append(s(t_h4s_lists_list(TV_u_27a),h4s_lists_nil),s(t_h4s_lists_list(TV_u_27a),V_l))) = s(t_h4s_lists_list(TV_u_27a),V_l)).
fof(ah4s_bools_IMPu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) => p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_sats_dcu_u_conj, axiom, ![V_r, V_q, V_p]: ((p(s(t_bool,V_p)) <=> (p(s(t_bool,V_q)) & p(s(t_bool,V_r)))) <=> ((p(s(t_bool,V_p)) | (~ (p(s(t_bool,V_q))) | ~ (p(s(t_bool,V_r))))) & ((p(s(t_bool,V_q)) | ~ (p(s(t_bool,V_p)))) & (p(s(t_bool,V_r)) | ~ (p(s(t_bool,V_p)))))))).
fof(ah4s_bools_DEu_u_MORGANu_u_THMu_c0, axiom, ![V_B, V_A]: (~ ((p(s(t_bool,V_A)) & p(s(t_bool,V_B)))) <=> (~ (p(s(t_bool,V_A))) | ~ (p(s(t_bool,V_B)))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c2, axiom, ![V_t]: ((p(s(t_bool,f)) | p(s(t_bool,V_t))) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c3, axiom, ![V_t]: (s(t_bool,V_t) = s(t_bool,f) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_bools_NOTu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_P]: (~ (?[V_x]: p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))) <=> ![V_x]: ~ (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_RESu_u_EXISTSu_u_THM, axiom, ![TV_u_27a]: ![V_f, V_P]: (p(s(t_bool,h4s_bools_resu_u_exists(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27a,t_bool),V_f)))) <=> ?[V_x]: (p(s(t_bool,h4s_bools_in(s(TV_u_27a,V_x),s(t_fun(TV_u_27a,t_bool),V_P)))) & p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_f),s(TV_u_27a,V_x))))))).
fof(ah4s_bools_ORu_u_CLAUSESu_c0, axiom, ![V_t]: ((p(s(t_bool,t)) | p(s(t_bool,V_t))) <=> p(s(t_bool,t)))).
fof(ah4s_bools_TYPEu_u_DEFINITIONu_u_THM, axiom, ![TV_u_27a,TV_u_27b]: ![V_rep, V_P]: (p(s(t_bool,h4s_bools_typeu_u_definition(s(t_fun(TV_u_27a,t_bool),V_P),s(t_fun(TV_u_27b,TV_u_27a),V_rep)))) <=> (![V_xu_27, V_xu_27u_27]: (s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27u_27))) => s(TV_u_27b,V_xu_27) = s(TV_u_27b,V_xu_27u_27)) & ![V_x]: (p(s(t_bool,happ(s(t_fun(TV_u_27a,t_bool),V_P),s(TV_u_27a,V_x)))) <=> ?[V_xu_27]: s(TV_u_27a,V_x) = s(TV_u_27a,happ(s(t_fun(TV_u_27b,TV_u_27a),V_rep),s(TV_u_27b,V_xu_27))))))).
fof(ah4s_bools_EQu_u_CLAUSESu_c0, axiom, ![V_t]: (s(t_bool,t) = s(t_bool,V_t) <=> p(s(t_bool,V_t)))).
fof(ah4s_bools_EQu_u_CLAUSESu_c2, axiom, ![V_t]: (s(t_bool,f) = s(t_bool,V_t) <=> ~ (p(s(t_bool,V_t))))).
fof(ah4s_options_NOTu_u_NONEu_u_SOME, axiom, ![TV_u_27a]: ![V_x]: ~ (s(t_h4s_options_option(TV_u_27a),h4s_options_none) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))))).
fof(ah4s_options_SOMEu_u_11, axiom, ![TV_u_27a]: ![V_y, V_x]: (s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_x))) = s(t_h4s_options_option(TV_u_27a),h4s_options_some(s(TV_u_27a,V_y))) <=> s(TV_u_27a,V_x) = s(TV_u_27a,V_y))).
fof(ah4s_bools_EQu_u_IMPu_u_THM, axiom, ![V_t2, V_t1]: (s(t_bool,V_t1) = s(t_bool,V_t2) <=> ((p(s(t_bool,V_t1)) => p(s(t_bool,V_t2))) & (p(s(t_bool,V_t2)) => p(s(t_bool,V_t1)))))).
fof(ah4s_primu_u_recs_numu_u_Axiom, axiom, ![TV_u_27a]: ![V_f, V_e]: ?[V_fn]: (s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_0))) = s(TV_u_27a,V_e) & ![V_n]: s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))))) = s(TV_u_27a,happ(s(t_fun(TV_u_27a,TV_u_27a),happ(s(t_fun(t_h4s_nums_num,t_fun(TV_u_27a,TV_u_27a)),V_f),s(t_h4s_nums_num,V_n))),s(TV_u_27a,happ(s(t_fun(t_h4s_nums_num,TV_u_27a),V_fn),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_Decodes_decu_u_bnumu_u_defu_c1, axiom, ![V_uu_4]: (![V_n, V_h, V_tu_27]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_4),s(t_h4s_nums_num,V_n))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_tu_27))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,V_h),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,h4s_nums_0))))),s(t_h4s_lists_list(t_bool),V_tu_27))))) => ![V_uu_3]: (![V_h, V_n]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_3),s(t_bool,V_h))),s(t_h4s_nums_num,V_n))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_4),s(t_h4s_nums_num,V_n))),s(t_bool,V_h))) => ![V_uu_2]: (![V_h, V_vu_27]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2),s(t_bool,V_h))),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_vu_27))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_vu_27),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_bool,t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_3),s(t_bool,V_h))))) => ![V_uu_1]: (![V_m, V_h, V_t]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_1),s(t_h4s_nums_num,V_m))),s(t_bool,V_h))),s(t_h4s_lists_list(t_bool),V_t))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_decodes_decu_u_bnum(s(t_h4s_nums_num,V_m),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2),s(t_bool,V_h))))) => ![V_uu_0]: (![V_m, V_h]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_0),s(t_h4s_nums_num,V_m))),s(t_bool,V_h))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_1),s(t_h4s_nums_num,V_m))),s(t_bool,V_h))) => ![V_m, V_l]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_decodes_decu_u_bnum(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_l),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))))),V_uu_0),s(t_h4s_nums_num,V_m))))))))))).
fof(ah4s_Decodes_decodeu_u_numu_u_total, axiom, ![V_uu_9]: (![V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_9),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_lists_list(t_bool),V_tu_270))))) => ![V_uu_8]: (![V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_8),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_9),s(t_h4s_nums_num,V_v))) => ![V_uu_7]: (![V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_7),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_8))) => ![V_uu_6]: (![V_v, V_tu_270]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_6),s(t_h4s_nums_num,V_v))),s(t_h4s_lists_list(t_bool),V_tu_270))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_v))),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))))),s(t_h4s_lists_list(t_bool),V_tu_270))))) => ![V_uu_5]: (![V_v]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_5),s(t_h4s_nums_num,V_v))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_6),s(t_h4s_nums_num,V_v))) => ![V_uu_4]: (![V_v1]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_4),s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_pairs_pairu_u_case(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),V_v1),s(t_fun(t_h4s_nums_num,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_5))) => ![V_uu_3]: (![V_v6, V_t]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_3),s(t_bool,V_v6))),s(t_h4s_lists_list(t_bool),V_t))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v6),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_some(s(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),h4s_pairs_u_2c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_lists_list(t_bool),V_t))))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))),s(t_h4s_lists_list(t_bool),V_t))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_4))))) => ![V_uu_2]: (![V_v6]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2),s(t_bool,V_v6))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_3),s(t_bool,V_v6))) => ![V_uu_1]: (![V_v2, V_tu_27]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),happ(s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_1),s(t_bool,V_v2))),s(t_h4s_lists_list(t_bool),V_tu_27))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_bools_cond(s(t_bool,V_v2),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_tu_27),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_2))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_optionu_u_case(s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))),s(t_h4s_lists_list(t_bool),V_tu_27))),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),V_uu_7))))) => ![V_uu_0]: (![V_v2]: s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_0),s(t_bool,V_v2))) = s(t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool)))),happ(s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_1),s(t_bool,V_v2))) => ![V_l]: s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_decodes_decodeu_u_num(s(t_fun(t_h4s_nums_num,t_bool),h4s_combins_k(s(t_bool,t))),s(t_h4s_lists_list(t_bool),V_l))) = s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_lists_listu_u_case(s(t_h4s_lists_list(t_bool),V_l),s(t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))),h4s_options_none),s(t_fun(t_bool,t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_nums_num,t_h4s_lists_list(t_bool))))),V_uu_0)))))))))))))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) <=> (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))) = s(t_h4s_nums_num,V_n) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_nums_0)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_EQ, axiom, ![V_p, V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_p) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))) | (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_p),s(t_h4s_nums_num,h4s_nums_0)))))))).
fof(ah4s_arithmetics_SUBu_u_RIGHTu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_DIVu_u_UNIQUE, axiom, ![V_q, V_n, V_k]: (?[V_r]: (s(t_h4s_nums_num,V_k) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_r))) & p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,V_r),s(t_h4s_nums_num,V_n))))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,V_k),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_arithmetics_SUBu_u_LEFTu_u_ADD, axiom, ![V_p, V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) = s(t_h4s_nums_num,h4s_bools_cond(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))),s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_NOTu_u_NUMu_u_EQ, axiom, ![V_n, V_m]: (~ (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,V_n)) <=> (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))),s(t_h4s_nums_num,V_n)))) | p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m))))))).
fof(ah4s_arithmetics_MULTu_u_DIV, axiom, ![V_q, V_n]: (p(s(t_bool,h4s_primu_u_recs_u_3c(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n)))) => s(t_h4s_nums_num,h4s_arithmetics_div(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_q),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,V_q))).
fof(ah4s_arithmetics_SUCu_u_ONEu_u_ADD, axiom, ![V_n]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n)))).
fof(ah4s_arithmetics_NOTu_u_LEQ, axiom, ![V_n, V_m]: (~ (p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))))) <=> p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,V_m)))))).
fof(ah4s_arithmetics_MULTu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) | s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_ADDu_u_SUB, axiom, ![V_c, V_a]: s(t_h4s_nums_num,h4s_arithmetics_u_2d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_a),s(t_h4s_nums_num,V_c))),s(t_h4s_nums_num,V_c))) = s(t_h4s_nums_num,V_a)).
fof(ah4s_arithmetics_EVENu_u_DOUBLE, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_even(s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit2(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_n))))))).
fof(ah4s_arithmetics_ZEROu_u_LESSu_u_EQ, axiom, ![V_n]: p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_nums_num,V_n))))).
fof(ah4s_arithmetics_LESSu_u_EQu_u_TRANS, axiom, ![V_p, V_n, V_m]: ((p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n)))) & p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p))))) => p(s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p)))))).
fof(ah4s_arithmetics_MULTu_u_CLAUSESu_c2, axiom, ![V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero))))),s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,V_m)).
fof(ah4s_arithmetics_MULTu_u_COMM, axiom, ![V_n, V_m]: s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_arithmetics_u_2a(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_m)))).
fof(ah4s_bools_BOOLu_u_EQu_u_DISTINCTu_c0, axiom, ~ (s(t_bool,t) = s(t_bool,f))).
fof(ah4s_numerals_numeralu_u_sucu_c0, axiom, s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,h4s_arithmetics_zero))) = s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))).
fof(ah4s_arithmetics_ADDu_u_MONOu_u_LESSu_u_EQ, axiom, ![V_p, V_n, V_m]: s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))),s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_p))))) = s(t_bool,h4s_arithmetics_u_3cu_3d(s(t_h4s_nums_num,V_n),s(t_h4s_nums_num,V_p)))).
fof(ah4s_arithmetics_ADDu_u_EQu_u_0, axiom, ![V_n, V_m]: (s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,V_n))) = s(t_h4s_nums_num,h4s_nums_0) <=> (s(t_h4s_nums_num,V_m) = s(t_h4s_nums_num,h4s_nums_0) & s(t_h4s_nums_num,V_n) = s(t_h4s_nums_num,h4s_nums_0)))).
fof(ah4s_arithmetics_ADD1, axiom, ![V_m]: s(t_h4s_nums_num,h4s_nums_suc(s(t_h4s_nums_num,V_m))) = s(t_h4s_nums_num,h4s_arithmetics_u_2b(s(t_h4s_nums_num,V_m),s(t_h4s_nums_num,h4s_arithmetics_numeral(s(t_h4s_nums_num,h4s_arithmetics_bit1(s(t_h4s_nums_num,h4s_arithmetics_zero)))))))).
fof(ch4s_EncodeVars_fixedu_u_widthu_u_unit, conjecture, ![V_p]: p(s(t_bool,h4s_encodevars_fixedu_u_width(s(t_h4s_nums_num,h4s_nums_0),s(t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_bool),t_h4s_pairs_prod(t_fun(t_h4s_ones_one,t_h4s_lists_list(t_bool)),t_fun(t_h4s_lists_list(t_bool),t_h4s_options_option(t_h4s_pairs_prod(t_h4s_ones_one,t_h4s_lists_list(t_bool)))))),h4s_coders_unitu_u_coder(s(t_fun(t_h4s_ones_one,t_bool),V_p))))))).
